Projective Geometric Algebra

Projective geometric algebra (PGA) refers to a class of mathematical models in which geometric objects, operations for combining them in various ways, and transformations that can be applied to them are all part of a single algebraic structure that is a specific type of Clifford algebra. This website deals with two types of algebras called rigid geometric algebra and conformal geometric algebra, named for the types of transformations they contain. Projective geometric algebra a natural fit for areas of computer science, especially computer graphics and robotics, that routinely use the mathematical concepts that PGA encompasses.

The rigid geometric algebra (RGA) adds one dimension to ordinary space and incorporates representations for Euclidean points, lines, and planes as well as operations for performing rotations, reflections, and translations. It completely subsumes conventional models that include homogeneous coordinates, Plücker coordinates, quaternions, and screw theory (which makes use of dual quaternions). The conformal geometric algebra (CGA) adds two dimensions to ordinary space and includes everything in the rigid geometric algebra. It also adds representations for round objects like circles and spheres as well as dilations, inversions, and other conformal transformations. Both types of algebra contain operations for easily joining lower-dimensional geometries to form higher-dimensional geometries, for calculating intersections among all of the geometries, and for projecting one kind of geometry onto another.

This page is a central resource containing all of the work on the subject of geometric algebra by Dr. Eric Lengyel. A large repository of mathematical reference materials can be found on the following two wikis:

• Rigid Geometric Algebra Wiki.

• Conformal Geometric Algebra Wiki.

A C++ library that implements much of this math is available under the MIT license on GitHub.

There is a Discord channel for discussion about topics from the book.

Exercises for PGA Illuminated will be posted within the next few days.

Projective Geometric Algebra Illuminated

Projective Geometric Algebra Illuminated is a book for engineers, mathematicians, scientists, software developers, and anyone else who wants to learn practical methods in geometric algebra. This book builds on new foundations and paints a more complete picture than any existing work. It focuses on applications, it makes comparisons with conventional methods, and it candidly points out cases in which geometric algebra is not better. Only a working knowledge of basic linear algebra is assumed.

Projective Geometric Algebra Illuminated

The 36×24 inch reference posters below contain a huge amount of information, including new research from 2020–2024.
Click on the images for PDF versions. High-quality prints of these posters are available in the U.S. at the links below each image.

Projective Geometric Algebra
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Conformal Geometric Algebra
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27D1 27C7

This website is an official silver sponsor of the symbols ⟑ and ⟇, which have Unicode values U+27D1 and U+27C7. These symbols denote the geometric product and geometric antiproduct in projective geometric algebra.